Problem: Simplify; express your answer in exponential form. Assume $z\neq 0, p\neq 0$. $\dfrac{{(z^{2}p^{-5})^{-2}}}{{(z^{5}p)^{3}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{2}p^{-5})^{-2} = (z^{2})^{-2}(p^{-5})^{-2}}$ On the left, we have ${z^{2}}$ to the exponent ${-2}$ . Now ${2 \times -2 = -4}$ , so ${(z^{2})^{-2} = z^{-4}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{2}p^{-5})^{-2}}}{{(z^{5}p)^{3}}} = \dfrac{{z^{-4}p^{10}}}{{z^{15}p^{3}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{-4}p^{10}}}{{z^{15}p^{3}}} = \dfrac{{z^{-4}}}{{z^{15}}} \cdot \dfrac{{p^{10}}}{{p^{3}}} = z^{{-4} - {15}} \cdot p^{{10} - {3}} = z^{-19}p^{7}$